# Seminars & Events for 2011-2012

##### A new formulation of the Gross-Zagier formula

In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.

##### A Kunneth formula in monopole Floer homology

We establish a Kunneth formula in monopole Floer homology, which describes how the groups associated to 3-manifolds behave under the operation of connected sum. The proof is based on a rigidity principal for Floer theories satisfying the surgery exact triangle. This is joint work with Tom Mrowka and Peter Ozsvath.

##### Curvature of random metrics

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics. This is joint work with I. Wigman and Y. Canzani

##### Stochastic Twist Maps and Symplectic Diffusions

I discuss two examples of random symplectic maps in this talk. As the first example consider a stochastic twist map that is defined to be a stationary ergodic twist map on a planar strip. As a natural question, I discuss the fixed point of such maps and address a Poincare-Birkhoff type theorem.

##### Characteristic polynomials of the hermitian Wigner and sample covariance matrices

We consider asymptotics of the correlation functions of characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$ and the hermitian sample covariance matrices $X_n=n^{-1}A_{m,n}^*A_{m,n}$. We use the integration over the Grassmann variables to obtain a convenient integral representation.

##### The Geometry of Triple Linking

Given a link $L$ in $R^3$ with just two components $X = \{x(s):s \epsilon S^1 \}$ and $Y = \{y(t): t\epsilon S^1\}$, their ** linking number** can be defined as the degree of the

**$f_L: S^1 \times S^1\rightarrow S^2$ , given by $f_L(s,t)=(y(t)-x(s))/|y(t)-x(s)|$, or as the value of the**

*Gauss map***, $1/4\pi\int_**

*Gauss integral*##### The mathematics of desertification: searching for early warning signals

The process of desertification can be modeled by systems of reaction-diffusion equations. Numerical simulations of these models agree remarkably well with field observations: both show that 'vegetation patterns'—i.e.

##### Bordered Floer homology

Bordered Floer homology is an invariant for three-manifolds with parameterized boundary. It associates to a parameterized surface a differential graded algebra, to a three-manifold with boundary a module over that algebra. It can be used to recapture the invariant of a closed three-manifold, via the derived tensor product. I will describe this construction and some of its applications.

##### Graph regularity and removal lemmas

Szemeredi's regularity lemma is one of the most powerful tools in graph theory. It was introduced by Szemeredi in his proof of the celebrated Erdos-Turan conjecture on long arithmetic progressions in dense subsets of the integers.

##### A combinatorial description of homotopy groups of spheres

The talk is based on the recent results obtained jointly with Jie Wu. For every $n>k>3$, we construct a group given by explicit generators and relations whose center is exactly the $n-th$ homotopy group of the k-sphere.

##### Toward a minus version of bordered Heegaard Floer homology

Bordered Heegaard Floer homology, in its current state, can recover the hat version of Heegaard Floer homology for closed manifolds. It is related to the sutured Floer homology, and gluing sutured manifolds along surfaces with boundary.

##### Subtle invariants and Traverso's conjectures for p-divisible groups

Let $D$ be a p-divisible group over an algebraically closed field $k$ of positive characteristic $p$. We will first define several subtle invariants of $D$ which have been introduced recently and which are crucial for any strong, refined classification of $D$. Then we will present our results on them.

##### The shooting method and the analysis of the target map via the degree theory

We introduce and analysis the ‘target map’ for the shooting method. For a large class of elliptic systems as well as more general dynamic systems, we show that the target map is onto via the degree theory. The target map is onto implies that we can shoot to any desired position.

##### BPS states, Donaldson-Thomas invariants and the Hitchin system

This talk will report on joint work with Wu-yen Chuang and Guang Pan relating the cohomology of the Hitchin system to Donaldson-Thomas theory and BPS inavariants of Calabi-Yau threefolds. A string-theoretic construction will be presented which relates refined curve-counting invariants to the work of Hausel and Rodriguez-Villegas on character varieties.

##### Orientability and open Gromov-Witten invariants

I will first discuss the orientability of the moduli spaces of J-holomorphic maps with Lagrangian boundary conditions. It is known that these spaces are not always orientable and I will explain what the obstruction depends on. Then, in the presence of an anti-symplectic involution on the target, I will give a construction of open Gromov-Witten disk invariants.

##### The Weil conjecture for singular curves

For a smooth curve, the Hilbert schemes of points are just symmetric powers of the curve, and their cohomology is easily computed in terms of H1 of the curve. In joint work with Davesh Maulik, we generalize this formula to curves with planar singularities (as conjectured by Migliorini and proved independently by Migliorini and Shende).

##### Local well-posedness of the KdV equation with almost periodic initial data

We prove the local well-posedness for the Cauchy problem of Korteweg-de Vries equation in an almost periodic function space. The function space contains functions satisfying $f=f_1+f_2+...+f_N$ where $f_j$ is in the Sobolev space of order $s>?1/2N$ of $a_j$ periodic functions. Note that f is not periodic when the ratio of periods $a_i/a_j$ is irrational.

##### Partial desingularization of pairs

Partial desingularization consists in removing all singularities, except for those of certain class $S$, with a proper birational map that is an isomorphism over the points already in $S$. For example, if $S$ consists only of the smooth singularities, then a partial desingularization in this sense corresponds to the usual (strong) resolution of singularities.

##### QED in Half Space

A proposal for QED in half space is made. Starting from the well known principle of mirror charges in electrostatics, we formulate boundary conditions for electromagnetic fields and charge carrying currents both in the classical and the quantum context. Free classical and quantum fields are constructed, such that the required boundary conditions hold. Conservation laws are discussed.

##### Partial desingularization of pairs

Partial desingularization consists in removing all singularities, except for those of certain class S, with a proper birational map that is an isomorphism over the points already in S. For example, if S consists only of the smooth singularities, then a partial desingularization in this sense corresponds to the usual (strong) resolution of singularities.